Consider the Cauchy problem of the Navier–Stokes equations in \(\mathbb {R}^n (n \ge 2)\) with the initial data \(a \in \dot{B}^{-1+n/p}_{p, \infty }\) for \(n< p < \infty \) . We establish the Gevrey type estimates for the error between the successive approximations \(\{u_j\}_{j=0}^{\infty }\) and the strong solution u provided the convergence in the scaling invariant norm in \(L^q(\mathbb {R}^n)\) with the time weight holds. It is also clarified that the convergence rate of the higher order approximation is at least the same as that of the lower order approximation. In addition, the approximation for the pressure is also established.
